I suggest to make it possible to set a timer, to close the private browsing session after an x time of inactivity. In the past there was an Add-in, which did, but this add-in is not available anymore.
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Still sounds like something an addon with the current addion api should be able to do, since it's mostly just closing a (private) window. Of course the addon would need to be enabled in private browsing mode ... to work, but thats a minor issue i guess.
I agree this feels like a feature only a minority of users will want - i.e an extension sounds relevant